Economic Growth with Trade in Factors of ProductionKarine Yenokyan*

Nazarbayev University

John J. SeaterNorth Carolina State University

Maryam ArabshahiChester�eld MO

December 2012

Abstract

We study the world trading equilibrium in a Ricardian model where factors of productionare themselves produced and tradable rather than endowed and non-tradable, corresponding tothe three-quarters of international trade that is in intermediate and capital goods. We showthat trade a¤ects economic growth purely through comparative advantage, even in the absenceof technology transfer, research and development, and international investment, and also in theabsence of aggregate scale e¤ects. Trade may raise a country�s growth rate or leave it unchanged.When a world balanced growth rate exists, trade always raises the growth rate of both tradingpartners. Otherwise, either one partner�s growth rate is increased and the other is una¤ected orneither partner�s growth rate is a¤ected, depending on the patterns of comparative and absoluteadvantage. Trade�s e¤ect on a country�s growth rate depends on the nature of the imported goodand not the exported good, a result contrary to the direction of many countries�export policies.Trade in factors of production e¤ectively transfers technology by producing an equilibrium identicalto that which would obtain if technology had been transferred between trading partners eventhough no such transfer actually occurs, which suggests that existing empirical evidence on therelation between trade and technology transfer may not be evidence that trade facilitates actualtechnology transfer. We show the conditions under which factor price equalization, the Stolper-Samuelson theorem, and the Rybczynski theorem hold. We perform a numerical analysis of thetransition dynamics, which appear to be saddle-point stable but may be monotonic or oscillatoryin converging to the balanced growth path.

Keywords: trade, growth, comparative advantage, world income distribution, e¤ective techno-logy transfer

JEL classi�cation codes: O4, F15

*We thank three very professional and helpful referees for comments that greatly improved thepaper and also Areendam Chanda, Robert Kane, and Pietro Peretto for earlier comments thathelped form the paper..

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1 Introduction

Does trade promote economic growth? Romer�s (1986) seminal article on endogenous growththeory provided economists with the proper framework for addressing that important question.Investigators soon started using that framework to study the growth e¤ects of trade. Among theearly contributions were Grossman and Helpman (1990, 1991), Rivera-Batiz and Romer (1991),and Young (1991). Later insights were provided by Taylor (1993), Feenstra (1996), Barro andSala-i-Martin (1997), Ventura (1997), and Redding (1999). Research on the topic has continuedup to the present in work such as Connolly (2000), Howitt (2000), Acemoglu and Ventura (2002),Galor and Mountford (2006), and Coe, Helpman, and Ho¤maister (2008). That literature�s answerto the question of whether trade promotes growth generally has been "Yes," with trade�s e¤ectworking through two channels: an aggregate scale e¤ect and technology transfer. The scale e¤ectarises from the increase in �rms�market size that opening to trade enables. A larger market sizemakes �rms more pro�table and so leads them to do more of the activities that cause economicgrowth. The technology transfer channel arises from trade�s facilitating knowledge spillovers ascountries set up lines of communication with their trading partners.

Surprisingly, the existing work actually leaves unanswered the question of whether trade itselfpromotes growth. The presence of aggregate scale e¤ects has been decisively rejected by the data,starting with the well-known article by Backus, Kehoe, and Kehoe (1992), so that .channel fortrade to in�uence growth can be dismissed, taking with it all the explanations of trade�s growthe¤ects that depend on the scale e¤ect. The second-generation fully endogenous growth literature(Peretto, 1998; Howitt, 1999) provides the theoretical reason for scale e¤ects to be absent. Thedata are much kinder to technology transfer facilitated by trade, which seems to be statisticallyand economically signi�cant (Coe and Helpman, 1995; Coe, Helpman, and Ho¤maister, 2008).However, in that mechanism it is not trade that a¤ects economic growth but the technologytransfer that trade facilitates. Without the technology transfer, trade would have no e¤ect ongrowth. Conversely, without trade, technology transfer still would a¤ect growth as long as someof it can occur independently of trade, which seems realistic. Interestingly, there also is evidencethat trade may a¤ect growth directly rather than through technology transfer. Alcala and Ciccone(2004), for example, �nd strong and robust e¤ects of trade on growth. Their measure of opennessto trade does not depend in any way on technology transfer and so suggests that trade in and ofitself boosts growth. This conclusion is buttressed by Wacziarg and Welch�s (2003) �nding thatincreased trade liberalization is positively associated with increased growth. Although these resultsare not decisive, they do suggest that trade per se a¤ects economic growth. What is missing is atheoretical argument showing why trade should have such an e¤ect.

The question we want to address, therefore, is precisely whether trade per se a¤ects growthin a model without empirically-rejected aggregate scale e¤ects. Surprisingly little literature hasexamined that question. Indeed, we are aware of a single article that uses the second-generationframework (which has no aggregate scale e¤ect) to study the interaction of trade and growth:Peretto (2003), In that model, the growth e¤ects of trade arise through knowledge spillovers,which are a passive form of technology transfer. Trade itself does not promote growth.

We present a model of trade and growth that has neither aggregate scale e¤ects nor technologytransfer. We show that trade, in and of itself, can raise growth through the same comparativeadvantage mechanism that raises the income level in static models without growth. The crucialelements are that growth is endogenous and that the factors of production are tradable. The tradepart of the model is Ricardian, with trade driven by cross-country technology di¤erences. Themodel di¤ers from standard Ricardian models in that factors of production are not endowed butrather are produced. The growth part of the model is the general two-sector model described byBarro and Sala-i-Martin (2004, chapter 5). There are two goods, both of which can be producedby each country. The two goods are produced in separate sectors. In the main analysis, eachgood can be used as a factor of production, and both goods are essential for production in both

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sectors. Allowing the factors of production to be produced permits endogenous growth. Allowingthe factors of production to be traded generates growth e¤ects of trade. The model is su¢ cientlytractable to allow analysis of the transition dynamics as well as the balanced growth path (hereafter,BGP).1

We obtain several interesting results. First, trade working solely through comparative advant-age can raise countries�growth rates. That is not to say, of course, that other channels, suchas technology transfer, are unimportant. What it does say is that the most important channelthrough which trade has economic e¤ects in traditional static models also operates on growth rates,something for which there has been surprisingly little theoretical support heretofore. Second whatmatters for the e¤ect of trade on a country�s growth rate is the type of good it imports, not thetype it exports. Speci�cally, importing a factor of production increases a country�s growth rate,whereas importing a consumption good has no e¤ect on the growth rate. The type of good thatis exported is irrelevant to the exporting country�s growth rate, irrespective of whether or not it isa factor of production. Third, trade can equalize countries�growth rates and therefore lead to astable distribution of GDPs across countries, but that is not a necessary outcome. Growth ratesmay remain permanently unequal after previously autarkic economies open to trade, leading to apermanently widening of the gap between their levels of GDP. The case of equal growth rates isthe same as Acemoglu and Ventura�s (2002) result, and it occurs when the world is in an interiorRicardian trade pattern, with each country completely specializing in producing one of the twofactors of production and trading to obtain the other factor. The case of unequal growth ratesoccurs when the world is in a corner Ricardian solution, in which one country does not specialize,and is a case Acemoglu and Ventura did not consider. The model thus o¤ers a generalization oftheirs that seems to correspond to some observed cases in the historical data. Fourth, trade infactors of production leads to a world equilibrium that is either identical or similar to the equilib-rium that would prevail if countries transferred technology to their partners, even though in themodel no technology transfer actually occurs. The identical case arises in the interior solution,and the similar case arises in the corner solution, mentioned in the previous result. Thus we havea sort of technology equalization result, similar to the factor price equalization theorem, accordingto which trade can e¤ectively equalize technology in whole or in part without any exchange of tech-nology. Fifth, we show that factor price equalization holds under conditions exactly opposite thoserequired in a Hecksher-Ohlin model with endowed factors and that neither the Stolper-Samuelsone¤ect nor the Rybczynski theorem hold in this kind of model. The analysis casts new light on factorprice equalization, showing that it arises when technology is the same across countries, either byassumption (as in Hecksher-Ohlin) or because trade makes it so (as in the Ricardian model usedhere). Sixth, our numerical analysis of the transition dynamics suggests that the balanced growthpath is saddle point stable and that the transition paths may be monotonic or oscillatory. Theendogenous nature of the world relative price is important in determining the transition dynamics,which are completely di¤erent from the dynamics of the two sector closed economy endogenousgrowth model, despite the fact that current model is based on the same structure as the closedmodel. Trade introduces important new elements.

2 Model Speci�cation

The analysis is based on an extension of the standard two-sector growth model discussed by Barroand Sala-i-Martin (2004, chapter 5) to allow trade between two countries. Indeed, the economicstructure of each of the two countries in our model is identical to that of the standard model. The

1 In all the literature on endogenous growth with international trade, only the pioneering but unfortunatelyunknown study by Bond and Trask (1997) relies on pure comparative advantage to generate e¤ects of trade ongrowth in a model without scale e¤ects or technology transfer. We compare their results with ours below.

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only new element is that we allow trade. Because the speci�cation is mostly standard, we consignall derivation and mathematical detail to the Appendix.2

2.1 Production and Preferences

Country i produces two goods, Y CKi and Y Hi . Good YCKi can be used for consumption C or

can be used as gross investment IK in one kind of capital K. Good Y H can be used only asgross investment IH in a second type of capital H produced in a di¤erent sector from Y CK by adi¤erent technology. Both Y CKi and Y Hi are tradeable. Each sector�s production technology isCobb-Douglas with country-speci�c parameter values:

Y CKit = Ai(vitKit)�i(uitHit)

1��i (1)

Y Hit = Bi[(1� vit)Kit]�i [(1� uit)Hit]

1��i (2)

where v and u are the fractions of total K and H used in the Y CK-producing sector and Ai, Bi,�i, and �i are constants. Both K and H are freely mobile between the two sectors, so v and ucan take any value in [0; 1] at any time t. Each of these production functions is homogeneousof degree 1 in K and H, thus satisfying the critical requirement for endogenous growth that themarginal products of the reproducible factors of production be bounded away from zero (Jonesand Manuelli, 1990).3 Both K and H depreciate at rate �, which is the same for both countries.The equations for the accumulation of K and H are

_Kit = Ai (vtKit)�i (utHt)

1��i � Ct � �Kit (3)

_Hit = Bi[(1� vit)Kit]�i [(1� uit)Hit]

1��i � �Hit

We can de�ne gross domestic product:

Yit = Y CKit + pY Hit (4)

where p is the price of Y H in terms of Y CK . Utility is CRRA, so lifetime utility is:

U =

Z 1

0

C1��it � 11� � e��tdt

2.2 Relation to Other Models

Endogenous growth models fall into four broad classes. The �rst class comprises R&D models ofvariety expansion or quality improvement. Those can be divided into three sub-classes: (1) the�rst-generation fully endogenous models of Grossman and Helpman (1991) or Barro and Sala-i-Martin (2004, chapters 6 and 7), and the like, (2) semi-endogenous growth models derived fromJones (1995), and (3) second-generation fully endogenous growth models of Peretto (1998), Howitt(1999), and their o¤spring. The second class comprises models based on learning-by-doing withknowledge spillovers, such as Romer (1986). The third class contains the two-sector models, suchas Barro and Sala-i-Martin (2004, chapter 5). The fourth class comprises models based on CESproduction with a high elasticity of substitution between a reproducible factor, such as capital,

2There are two mathematical appendices, one short and one long. The short appendix is included at the end ofthis paper and addresses only a few major points of the analysis. The long appendix is available from the authorsupon request and is more complete.

3This 2-sector model is not an AK model. AK models have no transition dynamics, whereas a two-sector modelsuch as this one has very rich dynamics. We discuss the transition dynamics below. See Chapter 5 in Barro andSala-i-Martin (2004) and Bond, Wang, and Yip (1996) discussions of the two-sector model�s dynamics for a closedeconomy.

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and non-reproducible factor, such as labor, as discussed in Barro and Sala-i-Martin (2004, chapter1). The AK model is a special case of the CES model with an in�nite elasticity of substitutionand a coe¢ cient on labor of zero.4

Even though there is no R&D in our model, we interpret our framework as an approximationto a second-generation fully endogenous R&D growth model. That interpretation is motivated bythe way we think of H. One common interpretation of H is as human capital, as in Uzawa (1965),Lucas (1988), or Barro and Sala-i-Martin (2004, chapter 5). However, another interpretation ofH that is more suitable for our purposes is technical progress embodied in physical capital thataugments labor. Gort, Greeenwood, and Rupert (1999) have shown that technical progress of thattype is important, accounting for about 52 percent of economic growth. Often technical progress istreated as augmenting the factor in which it is embodied, but embodiment and augmentation haveno necessary connection to one another. The factor that embodies a technology is not necessarilythe factor augmented by that technology. For example, consider a quilt maker using a traditionalsewing machine. Machine quilting requires a considerable amount of skill on the part of the quilter,who must move the fabric under the needle to produce the patterns of stitching. Quilting sewingmachines have recently become available that operate by moving the machine over the fabric, whichturns out to require far less skill on the part of the user. Thus a given quilt can be produced bya given person who either acquires a traditional machine and su¢ cient skill to use it or acquiresa quilting machine and does not bother with the skill. The technical progress embodied in thequilting machine acts exactly like skill embodied in the worker, and so such progress should enterthe production function in the same way: as a labor-augmenting reproducible factor - that is, asH - even though it is not embodied in the worker. Aghion and Howitt (2005) and Peretto (2007)present models in which technical progress augments labor but is embodied in goods. That is justthe framework we need for our analysis, but we simplify it because it is very complicated for amodel that also has trade in factors of production. We therefore outline the approach and thenpropose a simpli�ed version that is suitable for our purposes.

Peretto (2007) considers an economy in which �nal goods F are produced with a variety ofintermediate goods Ri and labor L. The production technology for a �nal good F is

F =

Z N

0

R�i�Z�i Z

1��Li�1��

di

where � and � are constants between zero and one, Ri is the quantity of intermediate good i, Nis the number of varieties of intermediate goods, Li is the quantity of labor using Ri, Zi is thequality of Ri, and Z is the average value of the Zi, given by

Z =1

N

Z N

0

Zjdj

Notice that the quality Zi is embodied in Ri but augments labor in �nal goods production. Theintermediate goods Ri are produced by monopolistic competitors, who also do R&D to raise thequality Zi of their product Ri. Increases in Zi raise the demand for Ri and thus also raise the priceof Ri, which in turn increases the monopoly pro�t of Ri�s producer. That (incipient) increase inpro�t induces new �rms to pay a sunk cost to enter the intermediate goods industry, raising N .

Peretto�s model provides many useful insights, but its complexity renders it di¢ cult to usein analyzing trade. It e¤ectively has three sectors - �nal goods, intermediate goods, and R&D -together with a mixture of perfect and imperfect competition, endogenous entry, and both varietyexpansion and quality improvement. We therefore simplify the model in the following ways.What we need for our subsequent work are two tradable goods that are factors of production,

4Ventura (1997) uses a CES model to discussion of how trade, through comparative advantage, distributes growthamong countries. In Ventura�s model, trade has no e¤ect on the world�s balanced growth rate (see his equation 11).

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one of which augments labor. We do not need an expanding variety of intermediate goods forwhat we are doing, so we �x the number of varieties at 2. The two varieties are types of physicalcapital. One corresponds to Peretto�s R and is standard physical capital that enters the productionfunction in the usual way. Converting a non-durable intermediate good into a durable capital goodalways is feasible, as Peretto remarks. The other capital good takes the place of Peretto�s Z, whichalready is a capital-like stock variable, and enters the production function as a labor-augmentingfactor. In the Aghion and Howitt and the Peretto frameworks, replacement of intermediate goodsby capital goods complicates the analysis, whereas in our case, it simpli�es it by allowing us to usethe symmetry inherent in the two-sector model. We also suppress labor so that we do not haveto deal with the issue of having more claimants on output than there is output to be distributed.Solving that problem requires introducing imperfect competition so that factors are not necessarilypaid their marginal product, but in a two-sector model imperfect competition is di¢ cult to dealwith. We thus treat H as a second type of capital that enters the production function in a waythat is complementary to K, giving constant returns to scale.

Several articles in the literature use the two-sector model to study trade and growth. See, forexample, Bond and Trask (1997), Bond, Trask, and Wang (2003), Farmer and Lahiri (2005, 2006),and Hu, Kemp, and Shimomura (2009). The distinguishing feature of our model is that we allowfor two tradable factors of production, not just one, something that has important implications forthe e¤ects of trade on growth.

3 Trade Between Two Large Countries

We now introduce foreign trade. We have two countries, 1 and 2, with di¤erent �xed productiontechnologies, discussed momentarily. In our framework of a two-factor Cobb-Douglas productionfunction, cross-country technology di¤erences are captured in di¤erent values of the total factorproductivities A and B and of the factor share parameters � and �. We assume that C and newunits of K and H are tradable but that the existing stocks of K and H are not tradable. In otherwords, investment goods (new units of K and H) are free to move about the world, but once theinvestment has been put in place, the resulting stocks of K and H are immovable. A factoryis an example. The materials to build the factory can be shipped abroad. Once the factory isassembled, however, it is immovable. We restrict attention here to sale of investment goods ofone type in exchange for investment goods of the other type, that is, exchange of new units ofK for new units of H. Under this assumption, a country that exports new units of K gives upownership of those units and accepts in return ownership of the new units of H that it receives asimports. We thus exclude net foreign investment, in which new units of capital are sent abroadbut ownership is retained. The reason is mathematical tractability. If we allowed net foreigninvestment, we would have four stocks of capital in each country - K and H owned by domesticresidents and K and H owned by foreigners. That would lead to eight state variables in the model,and the analysis would be totally intractable. By restricting attention to pure trade of K and H,we keep the number of state variables down to four, which is tractable.

It is traditional to begin (and often end) discussion of trade with the case of a �small�country,in which the country studied is an atomistic agent in the world economy. Here, however, itis preferable to follow Grossman and Helpman (1991, chapter 9) and analyze two countries ofarbitrary size. The small country is a straightforward special case of this more general frameworkthat we leave to the reader to work out.5 To avoid complications of bilateral monopoly, wesuppose that each country�s economy consists of a large number of competitive �rms with identical

5We do provide one major hint. With two large countries, the world price p responds to what both countries aredoing, and that response a¤ects the world�s dynamic path. With a small country, the world price does not respondto what the small country is doing.

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production functions, preventing either country from acting as a monopolist and thus guaranteeinga competitive solution.6

Let X denote exports of Y CK , so that p�1X is imports of Y H . Then the accumulation con-straints for country i are

_Kit = Ai(vitKit)�i(uitHit)

1��i +Xit � Cit � �Kit (5)

_Hit = Bi[(1� vit)Kit]�i [(1� uit)Hit]

1��i +1

pXit � �Hit (6)

When economies are closed, X = 0. When a country i is open, it must choose X along witheverything else. Countries 1 and 2 are linked through trade, so the solutions for their growth ratesmust be determined simultaneously. The key variable that guarantees world general equilibriumis the price p, which now is determined to guarantee international trade balance. Because neithercountry is small, equilibrium p depends on what both countries do, so that p, too, must be de-termined as part of the simultaneous solution for the two countries. We �nd the world generalequilibrium in two steps. First, we solve each country�s quasi-central planning problem, taking pas given; then we impose the trade balance condition X1 = �X2 to �nd the equilibrium value ofp7 .

It is important to note here that scale e¤ects, research and development, and technology transferall are absent from this model. Consequently, any e¤ects that trade has on growth will not be dueto those in�uences but rather to comparative advantage alone.

3.1 Individual Country Solutions

With trade, country i�s Hamiltonian is

Vi =C1��i

1� � e��t + �i

�Ai(viKi)

�i(uiHi)1��i � Ci � �Ki �Xi

�+

i

�Bi((1� vi)Ki)

�i((1� ui)Hi)1��i � �Hi +

Xi

p

�(7)

where � and are the costate variables. The control variables are C, X, v, and u. The importantnecessary condition for discussion here is the �rst-order condition for X:

@Vi@Xi

= ��i + ip= 0 (8)

The other necessary conditions are given in the Appendix.

The Hamiltonian is linear in X, so the �rst-order condition (8) for X does not depend onany control variable. We thus have bang-bang control for Xi. When ��i + ( i=p) is positive,equation (8) cannot be satis�ed; the marginal value of X equals ��i + ( i=p) and always ispositive, irrespective of the value X. Consequently, country i sets Xi as high as possible, whichit does by producing only Y CK and exporting some of it to obtain H. The opposite holds when��i + ( i=p) is negative. Country i then sets Xi as low as possible, producing only Y H andobtaining Y CK solely through imports (so that Xi is negative). When ��i + ( i=p) equals zero,country i does not engage in trade, and X is zero. To see this, note that ��i + ( i=p) = 0 is

6Analysis of the case where countries�national governments act as representatives for their countries��rms andbargain with other governments would be interesting but is beyond the scope of the present paper.

7Equivalently, we could solve for world general equilibrium as a world central planning problem, obtaining u1t,u2t, v1t, v2t, C1t, C2t, Xt, and pt in a single step. The two-step approach gives a more intuitive view of what eachcountry is doing.

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equivalent to p = i=�i. The price p is the international price for YH in terms of Y CK , that is,

the ratio of marginal utilities of Y H and Y CK . The costate variables �i and i are, respectively,country i�s marginal utilities of Y CK-goods and Y H -goods from internal production. Their ratiois the marginal value of Y H -goods in terms of Y CK-goods if country i produces both goods; thatis, the ratio is country i�s internal price for Y H in terms of Y CK . If this internal price equalsthe external (world) price p, country i can obtain the same number of units of Y CK in exchangefor Y H from its own internal operations as it can by trading on the world market. Country i isindi¤erent on the margin between trade and autarky. The borderline case of indi¤erence to tradewill prevail no more than momentarily because, as �i and i vary over time, they generally willnot satisfy the equality p = i=�i. Consequently, we ignore the knife-edge case henceforth.

Comparisons of the world price p with the internal prices i=�i are central to all that follows.The results of the previous paragraph suggest that country i specializes in producing Y H whenp > i=�i and in producing Y

CK when p < i=�i.

3.2 Balanced Growth Rates in World Equilibrium

We begin with the BGPs for the two countries and for the world as a whole. We address thetransition dynamics later.

3.2.1 Balanced Growth Paths under Autarky

The solution for the balanced growth rate under autarky is standard. The growth rates of C, K,H, Y CK , and Y are equal. Their common value is

=1

�

�A

�i1��i+�ii B

1��i1��i+�ii �

�i�i1��i+�ii (1� �i)

(1��i)�i1��i+�i �

(1��i)�i1��i+�ii (1� �i)

(1��i)(1��i)1��i+�i � � � �

�(9)

The value of p is

pi =

�Ai�

�ii (1� �i)1��i

Bi��ii (1� �i)1��i

� 11��i+�i

(10)

See the Appendix and the references cited there for details.

3.2.2 Balanced Growth Paths with Trade: Derivation

The equilibrium world price p must fall between 1=�1 and 2=�2; otherwise, both countrieswould try to specialize in and export the same good, violating international balance of trade.Which country has the higher value of i=�i is arbitrary, so we assume without loss of generalitythat 1=�1 > 2=�2. Since the ratio of costate variables represents the internal price for Y

H interms of Y CK , the ratio is equivalent to the autarkic price level in each country given by (10). Wethen have

p2 =

�A2�

�22 (1� �2)1��2

B2��22 (1� �2)1��2

� 11��2+�2

� p ��A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

= p1 (11)

For p to be strictly in the interior of the closed interval [p2; p1], country 1 sets v = u = 1, specializesin production of Y CK , and trades to obtain Y H . Country 2 does the opposite, producing onlyY H and trading for Y CK . When p is on the boundary of the interval, equaling either p1 or p2, wehave extra complications. For now, we restrict attention to the interior (i.e., the case where theinequalities in (11) are strict), discussing the corner cases afterward.

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The �rst-order conditions for C and X are unchanged from the unconditional problem, but the�rst-order condition for X now holds with equality. Having accepted the world price p and agreedto specialize in producing Y CK , country 1 now chooses � and to satisfy (8) exactly. The samekind of manipulations as for the autarkic model show that the growth rates of C, Y CK , K, andY all equal the growth rate of consumption. Because equation (8) now always holds, we havep = 1=�1. Trade balance constrains p to lie in the closed interval [p2; p1]. The growth rates forcountry 1 and 2 now are

1;T =1

�

"A1�

�11 (1� �1)1��1

�1

p

�1��1� � � �

#(12)

2;T =1

�

�B2�

�22 (1� �2)1��2p�2 � � � �

�(13)

where the subscript T indicates that this growth rate pertains when the countries trade.

Balanced growth requires that everything that grows must do so at a constant rate. The onlygrowth rate for p consistent with both these requirements is zero, so p must be constant along theBGP. The ratio 1=�1 therefore also is constant, implying that the growth rates of �1 and 1 areequal. It is straightforward to show that, for each country to have balanced growth individually,the two countries must have the same growth rate:

1;T = 2;T

With the two countries growing at the same rate, we have balanced growth for the world. Equating(12) and (13) and solving for p gives

p =

�A1�

�11 (1� �1)1��1

B2��22 (1� �2)1��2

� 11��1+�2

� (14)

Balanced growth thus requires that�A2�

�22 (1� �2)1��2

B2��22 (1� �2)1��2

� 11��2+�2

� ��A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

(15)

It is straightforward to show that

1;T R 2;T as p Q (16)

Substituting p = into (12) or (13) gives the common growth balanced growth rate

= 1;T = 2;T =1

�[�� � � �] (17)

where

� = ��1�2

1��1+�21 (1� �1)

(1��1)�21��1+�2 �

(1��1)�21��1+�22 (1� �2)

(1��1)(1��2)1��1+�2 A

�21��1+�21 B

1��11��1+�22

We discuss stability of the BGP below.

3.2.3 Balanced Growth Paths with Trade: Implications

The foregoing results lead to two interesting and important implications.

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Response of growth rates to trade In the interior solution that we are discussing here, wherethe world price p is inside the bounds given in (11), we obtain the important result that tradeincreases the growth rate of both trading partners. That is easily seen by comparing the growthrates under trade with those under autarky:

1;T =1

�

"A1�

�11 (1� �1)1��1

�1

p

�1��1#> 1;Au =

1

�

"A1�

�11 (1� �1)1��1

�1

p1

�1��1#

2;T =1

�

�B2�

�22 (1� �2)1��2p�2

�> 2;Au =

1

�

�B2�

�22 (1� �2)1��2p

�22

�The economic intuition is straightforward. In the interior trading equilibrium, each countryspecializes in the good of its comparative advantage and abandons production of the other good,obtaining it through trade instead of relatively ine¢ cient domestic production. From the world�spoint of view, productive resources in each country have been shifted from ine¢ cient to e¢ cientuses. Trade acts like technical progress, raising world Total Factor Productivity by leading toabandonment of the lower values of A and B. With higher TFP, growth rates rise. As we will seemomentarily, however, that result holds only in the interior. When the world price is on eitherboundary of the interval de�ned in (11), trade does not increase both countries� growth rates.However, it never lowers either partner�s growth rate.

E¤ective technology transfer Equation (17) shows that, in the interior, trade not only raisesboth countries�growth rates but also equalizes them. The economic intuition is that in the tradingequilibrium the two countries e¤ectively share each other�s relative e¢ ciency. For example, beforetrade, country 1 produces both Y CK and Y H . Domestic production of Y H uses the comparativelyine¢ cient technology of the domestic Y H sector. When trade starts, country 1 shuts down itsown Y H sector and relies instead on country 2�s relatively e¢ cient Y H sector. Country 2 is in asymmetric position, shutting down its Y CK sector and relying instead on its partner�s relativelye¢ cient Y CK sector. The result is that each country ends up relying on exactly the same mix of thetwo countries�technologies, leading to equal growth rates. In e¤ect, each country has transferredto itself the more e¢ cient technology of its trading partner, even though no technology transferactually occurs.

Another way to think of e¤ective technology transfer arises from our previous discussion ofthe interpretation of H. We argued that in our framework H is a proxy for technical progressembodied in tradable factors of production. In keeping with that view, we can think of trade infactors of production as a mechanism for transferring technology without the need for the recipientcountry to learn the production techniques of its trading partner. The recipient gets the bene�tof the technology embodied in the good it buys without having to learn the process for creatingthat technology.

E¤ective technology transfer through trade raises an interesting question for the interpretationof existing empirical work on technology spillovers. Coe and Helpman (1995), Coe, Helpman,and Ho¤maister (2008), and Keller and Yeaple (2009), among others, present evidence that tradein goods facilitates technology transfer. The argument is that trade makes it easier for tradingpartners to adopt each other�s technology. The foregoing results suggest another possibility, thattrade allows the partners not to import their partner�s technology but rather to substitute moree¢ cient production on their partner�s soil for their own less e¢ cient production on their own soil.The Coe-Helpman and Keller-Yeaple approach does not allow one to distinguish between the twomechanisms. Of course, both mechanisms could be at work. It is not immediately clear how tosort out the relative importance of the two channels, an issue left to future research.

9

3.3 Unbalanced Growth with Trade

As we have seen above, trade balance requires that the world price p falls in the closed interval[p2; p1] because otherwise the two countries would try to specialize in and export the same good.Balanced growth requires that p equals the quantity on the right side of (14). However, nothingguarantees that falls between p1 and p2. The only restriction we have imposed so far is thatp1 > p2 (in order to guarantee that trade occurs and to specify the direction of the trade �ows).That restriction puts no limits on the value of . We now analyze the e¤ect of trade when fallsoutside the critical interval.

3.3.1 Relation between growth rates

When falls outside the closed interval

[p2; p1] =

"�A2�

�22 (1� �2)1��2

B2��22 (1� �2)1��2

� 11��2+�2

;

�A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

#

the world price p cannot equal it because trade balance restricts p to be in the interval. Theworld price p then equals whichever interval boundary is closer to , and the world economy isat a corner. Recall that the endpoints of the interval are the internal, autarkic prices for the twocounties. When the world is at a corner solution, the growth rate of the country whose pricede�nes that corner is the same under trade as under autarky, the growth rate of the other countryis higher under trade than under autarky but lower than the growth rate of its trading partner,and balanced growth for the world and for the two countries individually is impossible.

Once again, results are symmetric for high and low values of , so without loss of generalityconsider the case where is larger than the upper boundary of the critical interval:

p1 =

�A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

<

The world price p then equals the upper boundary p1. Using that value for p in the growth rateformulae (12) and (13) gives the solutions

1;T = 1;A =1

�

8><>:A1��11 (1� �1)1��1"A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

# �(1��1)1��1+�1

� � � �

9>=>; 2;T =

1

�

8<:B2��22 (1� �2)1��2"A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

# �21��1+�1

� � � �

9=;The growth rate for country 1 is the same under trade as under autarky, the growth rate forcountry 2 is di¤erent than the autarkic rate, and the two growth rates clearly are di¤erent fromeach other.

The mathematical reason that country 1 grows at its autarkic rate even under trade is thatthe boundaries of the critical interval are the internal relative prices that would prevail underautarky, so when the world price hits the upper boundary, it equals the autarky price for country1. Substituting that price into the general growth rate formula (12) then returns the autarkygrowth rate. The economic intuition behind the mathematics is that at the corner country 1 doesnot specialize in producing just one good but instead produces both (as it must because its solutionis the same as under autarky). On the margin, it uses the same technologies under trade that ituses under autarky, so the growth rate under trade equals the growth rate under autarky.

10

The growth rate for country 2 is not the same as under autarky. Country 2 continues tospecialize in one good and trade for the other good. By abandoning production of one of the goodsand trading for it instead, country 2 continues to reap the e¢ ciency gains from trade which raiseits growth rate just as in an interior solution. As a result, country 2�s trade growth rate exceedsits autarky rate. To see that result formally, note that

2;A =1

�

(B2�

�22 (1� �2)1��2

�A2�

�22 (1� �2)1��2

B2��22 (1� �2)1��2

� �21��2+�2

� � � �)

<1

�

8<:B2��22 (1� �2)1��2"A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

# �21��1+�1

� � � �

9=;= 2;T

because by assumption�A2�

�22 (1� �2)1��2

B2��22 (1� �2)1��2

� 11��2+�2

<

�A1�

�11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

Even though country 2�s growth rate increases with trade, it remains below country 1�s growthrate. We can see that result from (16) together with p < .

In the corner case, then, trade leaves unchanged the growth rate of the country that does notspecialize and raises the growth rate of the country that specializes. The specialized country�sgrowth rate remains below the unspecialized country�s growth rate forever, so the corner solutionis stable. The specialized country�s production becomes a smaller and smaller fraction of worldoutput, which comes to be dominated by the unspecialized country. The unspecialized countrycontinues to import capital (either H or K, depending on whether p < or p > ) from thespecialized country but it also operates both sectors forever.8

To some extent, the reason that the world is in the corner solution with unbalanced growth isthat the specialized country�s total factor productivity is too low. The condition for the world tobe in the corner with country 2 specialized in the production of Y H is�

A1��11 (1� �1)1��1

B1��11 (1� �1)1��1

� 11��1+�1

< =

�A1�

�11 (1� �1)1��1

B2��22 (1� �2)1��2

� 11��1+�2

Some trivial algebra reduces this condition to the simpler expression

B2��22 (1� �2)1��2 < B1�

�11 (1� �1)1��1

If country 2 could increase its TFP parameter B2 in the Y H industry, it could reverse this inequalityand move the world out of the corner to the interior where world balanced growth with equal growthrates for all countries is possible. Of course, an increase in B2 alone would alter comparativeadvantages, but an equal increase in A2 and B2 would leave comparative advantage unchanged(see (11), the trade balance condition) and still move the world toward the interior. To that extent,we can say that unbalanced growth results from "excessively" low total factor productivities acrossall industries in one of the countries. The output/factor elasticities �i and �i also play a role in

8Bond, Trask, and Wang (2003) analyze a three-good, dynamic Hecksher-Ohlin model with two accumulatingfactors of production, one of which is not tradeable, and with all countries having identical production technolgies foreach good. Identical technologies make it possible for both countries to be incompletely specialized because tradeguarantees factor price equalization, as in the standard static Hecksher-Ohlin model. In contrast, in our modelincomplete specialization by both countries is not possible because di¤ering technolgies imply that factor prices arenot equalized whenever either country operates both sectors. See the discussion of factor price equalization below.

11

comparative advantage and the value of . The e¤ects of changes in � and � are of ambiguoussign, depending on whether � and � are greater or smaller than 1 in magnitude, so we cannot sayin general whether an increase in � or � would raise or lower the expressions for trade balance andfor . In contrast, the e¤ects of the TFP parameters Ai and Bi are unambiguous. We thus cansay that the country that falls behind the rest of the world (i.e., the specialized country) does soat least in part because its productivity is too low, an intuitively reasonable conclusion.

3.3.2 World income distribution

The possibility that trade does not equalize growth rates contrasts with Acemoglu and Ventura�s(2002) conclusion that trade leads to a stable world income distribution. Acemoglu and Venturapresent a model in which all countries converge to the same growth rate. Once growth rates areequal, relative incomes do not change, leading to their conclusion that the world income distributionstabilizes. The reason for the di¤erence between their results and ours is that they restrict theirmodel in such a way that it necessarily yields the equivalent of our interior solution, in which allgrowth rates are equal. In particular, they specify that each country is endowed with a monopolyin the production of a subset of intermediate goods, which no other country ever is permitted toproduce. Each country therefore is specialized from the outset by assumption. The specializationis imposed exogenously. Comparative advantage plays no role in determining it. Given thatexogenous �xed pattern of specialization, trade improves and equalizes all countries�growth ratesby allowing each country to use all intermediate goods that it cannot produce itself. The resultingequilibrium is mathematically equivalent to the interior solution of the model developed here, andcorner solutions are excluded a priori. In the less constrained analysis of the present model, incontrast, the pattern of production is determined endogenously by comparative advantage, andcorner solutions are possible outcomes. In the corners, world balanced growth does not occur, andthe world income distribution is not stable.

The possibility that growth rates fail to converge has practical value. Table 1, taken fromMaddison (2001), presents data on growth rates in various regions of the world going back athousand years. Over that entire time, Africa�s growth rate has lagged behind the rest of theworld. That is true even if one restricts attention to the more reliable data for the last 200 years.Our theoretical result of non-convergent growth rates o¤ers a possible explanation for the historicalbehavior of African growth rates.

3.3.3 E¤ective technology transfer again

When the world is in a corner, it is incompletely specialized. In that case, there still is e¤ectivetechnology transfer through trade but only in one direction: from the unspecialized country (theone in the corner) to the specialized country (the one not in the corner).

3.4 Trading factors and non-factors of production

The results so far have been derived for the case where both traded goods are factors of production.We now examine the case where one of the goods is not a factor of production, which leads to animportant conclusion concerning what it is about trade that can increase a country�s growth rate.

12

3.4.1 Interior and Corner Solutions

We suppose that Y CK-type goods (in the form of K) are not useful in production, only in con-sumption. That assumption requires that � = � = 0. The two production functions then are ofthe AK form:

Y CKi = AiuiHi

= Ci

Y Hi = Bi (1� ui)Hi

Going through the usual steps yields the autarkic growth rate for country i:

i;A =1

�(Bi � � � �)

Because Y CK is not a factor of production, its TFP parameter A has no e¤ect on the growth rate.

The condition (11) for trade balance simpli�es to

p2 =A2B2

� p � A1B1

= p1

Recall that we are assuming that country 1 has a comparative advantage in producing Y CK andcountry 2 has an advantage in good Y H . The balanced growth condition (15) becomes

A2B2

� A1B2

= � A1B1

(18)

With trade, country 1 specializes in Y CK , and country 2 specializes in Y H . The growth ratefor the interior solution is

= 1;T = 2;T =1

�[B2 � � � �]

The growth rate of both countries is determined only by TFP in the Y H -sector of country 2.Country 1�s TFP parameter A1 plays no role in the growth rate. When (18) is not satis�ed, theworld price p will be at one of the boundary values A1=B1 or A2=B2, and the world is in a cornersolution with one country specializing in production of one good and the other country producingboth goods and not specializing. When p equals A2=B2 > , country 1 produces only Y CK andimports Y H , and country 2 produces both Y CK and Y H and imports Y CK . The growth rates forthe two countries are

1;T =1

�

�A1

�B2A2

�� � � �

� 2;T =

1

�[B2 � � � �]

A little algebra shows that country 1�s growth rate 1;T is larger than under autarky, but it issmaller than country 2�s growth rate 2;T , consistent with relation (16) and p > . In this corner,then, trade has no e¤ect on the growth rate of country 2 (the country that does not specialize),raises the growth rate of country 1 (the country that does specialize), and leaves the specializedcountry�s growth rate below that of the unspecialized country.

Results are slightly di¤erent at the other corner. When p equals A1=B1 < , country 1 producesboth Y CK and Y H and imports Y H , and country 2 produces only Y H and imports Y CK . Thegrowth rates are the same as under autarky:

1;T =1

�[B1 � � � �]

13

2;T =1

�[B2 � � � �]

In this case, trade does not change either country�s growth rate. The mathematical constraintsplaced on the problem (i.e., A1=B2 > A1=B1 = p > A2=B2) imply that B1 > B2, so that 1;T > 2;T , again consistent with relation (16) because now p < .

3.4.2 Imports: The Driver of Trade-Enhanced Growth

These results make clear exactly how trade a¤ects growth of output and lead to an importantconclusion: What matters for output growth is not the good that is exported but rather the goodthat is imported. Growth rates depend on TFP in sectors that produce factors of production.Trade can raise a country�s output growth rate by allowing that country to substitute anothercountry�s higher TFP for its own. It is the importation of a factor of production that can raise acountry�s growth rate; it does not matter what good is exported in payment9 .

The economic intuition for this result goes to the heart of what makes perpetual growth possible.Ultimately, economic growth is driven by augmenting the non-reproducible factors of productionin such a way as to make the production function linearly homogeneous in the reproducible factorsof production. For example, in the Solow-Swan model, perpetual growth in income per personis possible if and only if technical progress is labor augmenting, as Phelps (1966) showed longago. In the standard model, physical capital and labor-augmenting technical progress are the tworeproducible factors of production, and the labor-augmenting nature of technical progress makes theproduction function linearly homogeneous in capital and technical progress. The crucial elementfor our purposes here is to recognize that it is the nature of the production function with respect tothe reproducible factors of production that makes perpetual growth possible. Similarly, trade canraise growth permanently if it makes the production of the reproducible factors of production moree¢ cient. The increase in e¢ ciency arises from comparative advantage, with each trading partnerspecializing in the factor in which it has a comparative advantage (i.e., in which it is the relativelye¢ cient producer). So, when each country stops producing one factor itself and instead obtains itby trading with its partner, it is making itself more e¢ cient in creating factors of production, andthat raises the growth rate. In contrast, importing a good that is not a factor of production doesnothing for increasing the e¢ ciency of making the goods that drive economic growth.

This result has an interesting policy implication. It is quite correct for countries to formulatetrade policies that promote production and export of the goods in which the country has a com-parative advantage. However, that is only half the necessary policy if the country seeks to increaseits growth rate. It also must ensure that at least some of the foreign exchange earned from theexports is used to import factors of production in which the country does not have a comparativeadvantage. Importing only consumption goods will do nothing for growth.

3.4.3 Pareto E¢ ciency of Trade with Respect to Growth Rates

Looking back over the various results we have obtained for the e¤ect of trade on growth, we see thattrade never reduces growth and in all but one case raises the growth rate of at least one tradingpartner. Thus - in this model, at least - trade�s e¤ect on growth is guaranteed to be weakly Paretoe¢ cient (not hurting any growth rate) and likely to be strongly Pareto e¢ cient (raising at least

9Bond and Trask (1997) have a three-sector model with separate sectors for C, K, and H. However, their analyisis substantially di¤erent from ours because they restrict attention to a single small open economy, assume that His human capital and non-tradeable, and do not analyze world equilibrium. The existence of only one tradeablefactor of production makes Bond and Trask�s growth implications a special case of the present model, essentiallya combination of special cases considered here. The presence of a non-tradeable factor of production guaranteesthat growth rates will not be equalized across countries unless the technology for producing that factor is the sameacross countries.

14

one growth rate without lowering any growth rate). That conclusion obviously has the strongimplication for the policy debate that opening to trade is bene�cial because it will not hurt growthand may help it.

3.5 Factor Price Equalization, Stolper-Samuelson, and Rybczynski

The famous theorems from standard trade theory assume the standard static Hecksher-Ohlin frame-work of two small open economies producing two goods with identical technologies but di¤erentfactor endowments. In contrast, the present model assumes a dynamic Ricardian framework inwhich two large open countries have di¤erent technologies and in which the factors of productionare endogenous rather than endowed. We now see to what extent the "big three" theorems ofclassical trade theory carry over to the present framework. We might expect at least some modi-�cation because the framework of analysis is so di¤erent from that underlying the conditions underwhich the theorems originally were derived. Indeed, Jones (1965, p.563) shows that his "magni-�cation e¤ect" need not hold in a general equilibrium framework where just �nal goods prices areendogenous, even taking the quantities of the factors of production as given. We now show thatthe endogeneity of the factors of production and the requirements of capital market equilibriumintroduce considerations absent from the static Hecksher-Ohlin framework and alter the theorems.Our results also provide a new perspective on the nature of the Factor Price Equalization theorem.

3.5.1 Factor Price Equalization

The dynamic Ricardian model has interesting implications for factor price equalization and revealsthe underlying reason for it.

In the present model, when the world is at an interior solution, country 1 operates only theY CK sector and country 2 only the Y H sector. Thus

Y CK1 = A1K�11 H1��1

1

Y H2 = B2K�22 H

1��22

From these equations it is straightforward to derive the marginal products of K and H in eachcountry, which are the gross rates of return to each type of capital, rKi and rHi, where i = 1; 2.Comparing marginal products across countries, we see that we have factor price equalization:

rK1 = prK2

rH1 = prH2

The situation is di¤erent in the corner solution. There, country 1 operates both sectors, andcountry 2 operates only the Y H sector. The production functions are

Y CK1 = A1

�KY CK

1

��1 �HY CK

1

�1��11

Y H1 = B1�KH1

��1 �HH1

�1��1Y H2 = B2K

�22 H

1��22

Examining marginal products shows that

rK1 6= prK2

rH1 6= prH2

so that factor price equalization does not hold in the corner. These results are interesting for atleast two reasons.

15

First, it might seem that trading the factors is su¢ cient to guarantee equal factor prices (asmeasured by rates of return). The corner solution shows that not to be the case. There, thetwo factors are traded, but they have di¤erent rates of return. The reason is that country 1obtains its marginal unit of H from itself, not from country 2, and country 1 is disadvantaged inthe production of H. The constraint is binding in the corner, and equality of marginal rates ofreturn does not hold.

Second, the conditions that yield factor price equalization in this dynamic Ricardian model areexactly the opposite of those required for the standard static Hecksher-Ohlin model. In the staticH-O framework, factor price equalization requires that both countries operate in the interior oftheir cones of diversi�cation, which means that both countries produce both goods, that is, neithercountry is specialized. In contrast, in this dynamic Ricardian model factor price equalizationrequires that the world be in the interior region where each country produces only one good, thatis, both countries are specialized. The contrasting conditions for factor price equalization in factrely on the same underlying phenomenon. In both types of models, factor price equalizationrequires that the two countries use the same technology. In the H-O framework, that requires thatboth countries produce both goods so that the relevant cost functions can take on the same value(see Feenstra, 2004). In this dynamic Ricardian framework, technology is e¤ectively equalizedby trade, but only for an interior solution. It is only in the interior that trade generates afull e¤ective technology transfer that leaves the two countries producing as if they actually hadexchanged technology. Thus we see that the deep underlying condition for trade to bring aboutfactor price equalization is that it must lead to an e¤ective equalization of technology. Once thathappens, international trade provides the linkages necessary for the market to reallocate factorsof production until they earn equal rates of return across countries. Without equalization oftechnology, as in the corner solution in either the H-O or Ricardian model, rates of return cannotbe equalized.

3.5.2 Stolper-Samuelson

The Stolper-Samuelson theorem shows that in the standard H-O framework an exogenous increasein the relative price of a good increases the return to the factor of production used intensively inthat good and decreases the return to the other factor. There are no exogenous changes in pricesin the present model because neither country is "small" in the sense of taking prices as given,so we cannot ask the question that the Stolper-Samuelson theorem addresses. We can ask tworelated questions, though: How do the returns to the factors of production respond to a changein an underlying parameter that causes a change in a relative price, and how much of the totale¤ect works through the change in price? The easiest parameters to study are the total factorproductivities Ai and Bi, so we start with them. Their e¤ects are identical except for sign, so werestrict attention to an increase in A1.

In the interior, both countries are specialized, so we know immediately that the Stolper-Samuelson theorem does not apply. We cannot ask the question it addresses, namely, the e¤ectsof a change in the relative price of two goods produced within a country, but we can ask about thee¤ect of a change in the world relative price of the two traded goods on the factor returns in each

16

country. The gross returns to K and H in each country are

rK1 = �1A1 (K1=H1)�1�1 = �1A1

�p

�11� �1

��1�1(19)

rH1 = prK1 = �1A1p�1

��1

1� �1

��1�1(20)

rK2 = �2B2 (K2=H2)�2�1 = �2B2

�p

�21� �2

��2�1(21)

rH2 = prK2 = �2B2p�2

��2

1� �2

�(22)

From (14) we see that p is a positive function of A1, so an increase in A1 has a direct e¤ect on rK1and rK2 and an indirect e¤ect (through p) on all four rates of return. The indirect e¤ect is whatinterests us here, and it is negative for rK1 and rK2 and positive for rH1 and rH2, irrespective ofwhich good is intensive in which factor (that is, the relation between �1 and �2). The economicinterpretation is straightforward. An increase in TFP in the Y CK industry induces an increasein the ratio of K to H, which necessarily reduces the marginal product of K and increases themarginal product of H. In the interior, there is nothing like the Stolper-Samuelson e¤ect at play.

The economy�s full response to a change in A1 (the direct and indirect e¤ects) is to increase allfour rates of return. Domestic capital market equilibrium requires that the two types of capitalhave the same rate of return, expressed in common units:

rK1 =rH1p

rK2 =rH2p

and international capital market equilibrium requires that each type of capital has the same returnacross countries:

rK1 = prK2

rH1 = prH2

The full solution for rK1 is

rK1 =

"A�21 �

�1�21 (1� �1)�2(1��1)

B�1�12 ��2(�1�1)2 (1� �2)

(1��2)(�1�1)

# 11��1+�2

which is a positive function of both TFP parameters A1 and B2. The intuition is straightforward.An increase in either TFP parameter raises the marginal products of both types of capital in thatproduction function. Capital market equilibrium then requires that the marginal products in theother country equal those in the �rst country. In contrast to Stolper-Samuelson, rates of returnmove in the same, not opposite, directions.

In the interior, specialization renders the simple, original version of the Stolper-Samuelsone¤ect inapplicable because each country produces only one good. Also, the endogeneity of K andH means that the factor "endowments" themselves respond to any change in the world relativeprice p of the two goods rather than remaining constant as in the standard Hecksher-Ohlin model,so that the K=H ratios in the two countries adjust to satisfy the requirements of capital marketequilibrium.

In the corner, one country does not specialize. As usual, consider the corner where it is country1 that is unspecialized. For country 1, we can ask the Stolper-Samuelson question of how a change

17

in the relative price of the two goods change relative rates of return. The rates of return for Kand H are the same as in (19) and (20) except that p is replaced by p1, given by (11). Herewe assume that the increase in A1 is not large enough to move the world out of the corner. Anincrease in A1 a¤ects the rates of return both directly and indirectly, just as in the interior. TheStolper-Samuelson e¤ect is the indirect e¤ect coming through the change in p1. An increase inA1 raises p1 (i.e., raises the relative price of H in terms of K). That reduces the rate of return toK and raises the rate of return to H, irrespective of the factor intensities in production. So onceagain the Stolper-Samuelson e¤ect is absent here. As in the interior, the full e¤ect of an increasein A1 is to raise both rates of return rK1 and rH1.

3.5.3 Rybczynski

The Rybczynski theorem addresses the e¤ect of an exogenous change in relative factor endowmentsand concludes that an increase in one of the factors of production will lead to an increase in outputin the industry that uses that factor intensively and a decrease in the output of the other industry.In the present model, no such e¤ect emerges. Instead, an increase in one factor leads to a decreasein the output of the industry that produces that factor and an increase in the output of the industrythat produces the other factor. The initial shock may shut down trade, depending on the initialpattern of trade.

Suppose the world is in the interior and on the BGP when a natural disaster reduces country1�s stock of K. The reduction in K reduces the current K=H ratio but does not change theratio�s equilibrium value. Because of the bang-bang nature of investment, country 1 responds byshutting down investment in H and devoting all investment to K until the K=H ratio is restoredto its equilibrium value. Under the assumptions we have been using, country 1 was exporting Kand importing H before the shock, something it no longer wants to do. Country 1 therefore alsoshuts down trade until it restores the K=H ratio to the equilibrium value. None of this responsedepends on which industry uses K intensively, so the Rybczynski theorem does not hold. Thesame conclusion holds by similar logic if the world starts in the corner solution when the shockoccurs.

The reason the Rybczynski theorem does not hold is that the factors of production are notendowments but rather are produced that countries want to hold in a ratio that depends on theunderlying parameters of the economy. A shock to the existing stock of a reproducible factor doesnot change the desired factor ratio, so the response to a shock it to increase production of thefactor in relatively short supply and shut down investment in the other factor altogether.

4 Transition Dynamics

We now turn to a study of the model�s transition dynamics.

Most of the existing literature on the transition dynamics of the two-sector model mostly studiesonly closed economies. Important contributions include Mulligan and Sala-i-Martin (1993), Bond,Wang and Yip (1996), Faig (1995), and Mino (1996). Mulligan and Sala-i-Martin considered ageneral model that exhibits constant returns to scale at the private level but also incorporatesincreasing or decreasing returns at the social level. The analysis of the transitional dynamics ofthe model is based on numerical simulations. Bond, Wang and Yip derived transitional dynamicsof a closed general two sector endogenous growth model. The dynamic properties of the model areanalyzed using the system consisting of three di¤erential equations in price (p), consumption perunit of human capital (c) and the ratio of physical capital to human capital (k). As they show thetransitional dynamics depend on the assumptions about the relationship between factor intensity

18

parameters. The asymptotic adjustment towards steady state can be driven by either adjustmentsin physical capital or by the stabilizing forces of the price level. Faig develops graphical tools toanalyze dynamic properties of the model with physical and human capital and then uses a derivedframework to analyze e¤ects of �scal policies and stochastic shocks. Mino focuses on the analyticalframework to analyze the dynamics of the two-sector model with physical and human capital inthe presence of capital income taxation. His arguments are in the same line with Bond, Wang andYip that dynamics of the economy and e¤ects of capital income taxation depend on assumptionsregarding the relative factor intensities in both sectors of production. These studies all contributesigni�cantly to the understanding of the transitional dynamics of two-sector endogenous growthmodels of closed economies.

Introducing trade into the two-sector model adds new dimensions to the transition dynamics.As Mino (1996) remarks, �Since the literature on sectoral shifts has usually ignored the possibilityof endogenous growth, the open-economy version may provide interesting contributions to the�eld� (page 247). Indeed, Ventura (1997) argues that the East Asian growth miracle can beexplained by �structural transformation�when faster accumulation of capital leads to the expansionof the capital-intensive sector and contraction of the labor-intensive sector and not just continuingproduction of both goods with more capital-intensive techniques. To the best of our knowledge, theonly contribution in the literature that follows Mino�s suggestion is that of Bond and Trask (1997).They analyze a small open economy with three sectors: capital, consumption, and education.Capital goods and consumption goods are tradable, and education is not tradable. They showthat the BGP for such an economy is saddle-point stable.

Our model di¤ers from that of Bond and Trask in three ways that a¤ect the economy�s transitiondynamics. First, in general both factors of production are tradable in our model. Second, we donot restrict attention to a small open economy but rather consider countries that may be large.Third, we analyze world general equilibrium. These key di¤erences lead to a substantially richerpattern of transition dynamics than emerge in the closed-economy case and that are consistentwith Ventura�s argument of �structural transformation.�

Our analysis is based on the general case where � 6= � and examines the transitional dynamicsin the neighborhood of the BGP with complete specialization, where country 1 specializes in theproduction of good Y CK and country 2 specializes in production of good Y H . An interestingaspect of the transition dynamics is that their properties depend heavily on the initial deviationsof factor ratios in both countries from their BGP values. The BGP solutions for the two countries�factor ratios k�1 � K1=H1 and k�2 � K2=H2 are

k�1 =�1

1� �1p (23)

k�2 =�2

1� �2p (24)

where as usual p is the world relative price determined by equation (14). The transition dynamicsabout the steady state are di¢ cult even with linearization because we have a �ve-dimensional sys-tem consisting of the dynamic equations for k1, k2, c1 = C1=K1, c2 = C2=K2, and p. Nonetheless,we can deduce some analytical results and obtain others numerically. There are four cases toconsider, depending on how the initial values of the factor ratios di¤er from their BGP values:

1. k1 < k�1 , k2 > k�2

2. k1 < k�1 , k2 < k�2

3. k1 > k�1 , k2 < k�2

4. k1 > k�1 , k2 > k�2

19

The �rst two cases are fairly easy because no trade occurs along the transition. That means thecountries�dynamics are independent of each other, allowing us to treat the two countries separatelyand obtain an analytical solution for the transition dynamics. The remaining two cases are morecomplicated because countries trade along the transition path. The countries are large relativeto each other, so they a¤ect the world relative price. Changes in the world price transmit thee¤ects of one country�s actions to the other country, requiring a simultaneous solution for the twocountries. The high dimensionality of the problem requires a numerical solution.

The main conclusions from the following analysis are that the BGP is saddle-path stable andthat transition dynamics may exhibit oscillatory behavior.

4.1 Case 1: k1 < k�1, k2 > k�2

In this case country 1 specializes in production of good Y CK on the BGP but deviates from theBGP value of its K=H ratio by having too much H capital (the good that it does not produce onthe BGP) relative to K. Country 2 is in the opposite situation with too much K and too little Hcompared to the BGP ratio. Because country 1 has relatively more of the H than its BGP value,it sets investment in H to 0. With investment in H equal to 0, country 1 does not import Y H

from country 2 along the transition to the BGP. This conclusion follows from the accumulationcondition for H in country 1:

_H1 + �H1 =X1

p= 0

which implies that H depreciates at rate �. Intuitively, country 1 has too much H, so the relativeprice of H in country 1 is very low. Country 1 no longer has a comparative advantage in theproduction of Y CK and so does not have an incentive to trade Y CK for Y H . With trade shutdown, the optimization problem faced by country 1 reduces to that of the closed economy one-sectorendogenous growth model. The present value Hamiltonian for country 1 becomes

V =C1��1 � 11� � e��t + �1

�A1K

�11 H1��1

1 � C1 � �K1

�(25)

The dynamics of that model are discussed in Barro and Sala-i-Martin (chapter 5, 2004). Thedynamic system for country 1 is written in terms of the variables c1 = C1=K1 and k1 = K1=H1

and expressed in terms of the following two dimensional system:�_c1=c1_k1=k1

�=

"1

(�1�1)(���1)A1k�1�21

�

�1 A1(�1 � 1)k�1�21

# �c1 � c�1k1 � k�1

�The phase diagram for country 1 in this case is similar to that for the closed economy one sectormodel in Barro and Sala-i-Martin (chapter 5, 2004) and is shown in Figure 1. Along the transitionpath country 1 depreciates H capital, accumulates K capital, and decreases consumption until itreaches the BGP value of k�1 , at which point country 1 opens to trade again.

Now, consider country 2. On the BGP country 2 specializes in producing Y H and importsY CK from country 1. The imported Y CK is used for consumption and investment in K. As wehave just discussed, country 1 no longer has an incentive to trade, so country 2 must open a sectorproducing good Y CK in order to have consumption. Country 2 also acts as a closed economy twosector model until the price level in country 1 reaches the level consistent with the BGP in thepresence of trade. At that price level both countries will have incentives to trade, so country 2 willresume specializing in Y H and trading to obtain Y CK from country 1.

The relative price for Y H in country 1 is determined by the ratio of marginal products of thetwo types of capital:

p1 =

�1� �1�1

�k1 (26)

20

It follows that the dynamics of the price along the transition path are determined by the accumu-lation conditions for K and H. The price dynamics in the two countries are shown in Figure 2. Aswe have already discussed the balanced trade condition requires world relative price to fall insidethe closed interval [p2; p1]. It follows from (26) that under the conditions of case 1 the relativeprice of H capital in country 1 will be lower than the world relative price level in the presence oftrade. However, as country 1 accumulates K capital and depreciates H capital along the transitionpath, the relative price level of good Y H increases until it reaches the world relative price levelconsistent with the presence of trade along the BGP. When the price in country 1 reaches the levelof p determined by equation (14) both countries will open to trade.

4.2 Case 2: k1 < k�1, k2 < k�2

Once again, country 1 has too much H relative to K, so the solution for country 1 is exactly thesame as in case 1. However, for country 2 the situation is di¤erent from the previous case. Now,country 2 also has too much H relative to K, so it, too, has no incentive to accumulate H. Country2 sets investment in H equal to 0 and accumulates only K. The present value Hamiltonian forcountry 2 is

V =C1��2 � 11� � e��t + �2

�A2K

�22 H1��2

2 � C2 � �K2

�(27)

This Hamiltonian looks exactly like its counterpart for country 1 given by (25), so the dynamicbehavior of country 2 will be similar to that of country 1. The two countries have qualitativelysimilar phase diagrams that are like Figure 1.

In case 2 both countries start with a high level of H relative to K. Therefore, in both countriesthe relative price of good Y H is lower than the expression for the world price level p given byequation (14) that would prevail on the BGP in the presence of trade.. It also can be shown thatthe starting relative price level in country 2 is lower than its autarkic price level associated withthe operation of both sectors.10 As the two countries accumulate K and depreciate H along thetransitional path, the relative price level in both countries increases. As explained earlier, theBGP world relative price level in the presence of trade falls inside the closed interval given by theautarkic BGP price levels in the two countries, with the upper and lower bounds of the intervalgiven by the autarkic price levels in country 1 and country 2, respectively. That means that, asthe relative price increases in both countries along the transition path, country 2 will achieve itsautarkic BGP price level �rst. At that point, country 2 will begin operating both sectors and willstay in autarky until the price level in country 1 reaches p. When that happens, trade will begin,and each country will specialize in the good for which it has a comparative advantage. Figure 3shows the price adjustment paths.

4.3 Case 3: k1 > k�1, k2 < k�2

We now suppose that country 1 has too much K relative to H and country 2 has too much Hrelative to K. The important property of this case is that at the starting point each country hasmore of the good for which it has a comparative advantage on the BGP. Because country 1 hasrelatively more K than H, it sets investment in K to 0 and exports its entire production of K:

X1 = A1K�11 H1��1

1 � C1 (28)

The domestic stock of K depreciates at rate �:

K1t = K10e��t (29)

10See Yenokyan (2010) for the details

21

Accumulation of H in the presence of complete specialization is given by

_H1

H1=A1k

�11 � c1k1p

� � (30)

The usual derivation shows that the growth rate of consumption in country 1 is

_C1C1

=1

�

�(1� �1)A1k�11

p+_p

p� � � �

�(31)

Equations (29), (30), and (31) determine the paths of C1, H1 and K1.

A similar analysis for country 2 gives the following three equations for determining the pathsof C2, H2 and K2:

_K2

K2= pB2k

�2�12 � c2 � � (32)

H2t = H20e��t (33)

_C2C2

=1

�

�p�2B2k

�2�12 � � � �

�(34)

To complete the solution, we need to determine the growth rate of the world relative price, p.The balanced trade condition is

A1H1k�11 � C1 = pB2H2k

�22

Total di¤erentiation of that condition and some algebra provide the growth rate of the worldrelative price:

_p

p=

�

(�pB2k�22 + c1k12)

A1k(�1�1)1 k12

�(1� �1)(A1k�11 � c1k1)

p� ��

� �

(�pB2k�22 + c1k12)

� pB2k�22h�2pB2k

�2�12 � �2c2 � �

i� �

(�pB2k�22 + c1k12)

� c1k12�

�(1� �1)A1k�11

p� � � �

�where k12 = K1=H2 is constant along the transition to the BGP because investment in both K1

and H2 is zero during the transition and K and H both depreciate at rate �.11

The transition behavior of this world economy can be expressed in terms of the following�ve variables p, c1 = C1=K1, k1 = K1=H1, c2 = C2=K2, k2 = K2=H2. The corresponding �vedimensional system is:266664

_p=p_c1=c1_k1=k1_c2=c2_k2=k2

377775 =266664n11 n12 n13 n14 n15n21 n22 n23 n24 n25n31 n32 n33 0 0n41 0 0 n44 n45n51 0 0 n54 n55

377775266664

p� p�c1 � c�1k1 � k�1c2 � c�2k2 � k�2

377775where the nij are various combinations of the parameters �1, �2, A1, B2, �, �, �, the BGP values ofk1and k2, and the initial value k12 (which is constant on the transition path).12 . The system canbe written as _z = Nzt, where _z is the �ve-dimensional vector of the growth rates of the variables,

11 see Yenokyan (2010) for the details of derivations12See Yenokyan (2010) for the expressions for the nij .

22

p, c1, k1, c2 and k2 and zt is a vector of deviations of the variables from their steady state values.The system�s solution can be approximated as:

zt =Me��tM�1z0

where M is the matrix of the eigenvectors of matrix N, � is the diagonal matrix of eigenvalues ofmatrix N and z0 is the vector of initial deviations of the variables from their steady state.

The high dimensionality of the above system does not allow an analytical solution, so wecalibrated the model and performed some simulations to study the system�s behavior. We presentonly a brief summary of the simulation results here. We set the time unit to a quarter of a year.To start the simulation exercise we imposed the values: �1 = 0:3, �2 = 0:25, � = 0:0025, � = 0:025,� = 2, and A1 = 1. The value of �1 is the usual capital share in a Cobb-Douglas productionfunction, taken as an average from the National Income and Product Accounts. The choice of theinitial value for �2 is based on the assumption that �1 > �2, meaning that the share of physicalcapital in the production of a labor-augmenting type of capital is smaller than in the productionof physical capital itself. This is the usual assumption that a factor�s share in its own productionis higher than in the production of other factors, or in other words that we do not have factorintensity reversal. With quarterly time units, the value of � = 0:0025 implies that the annual realinterest rate is 1 percent (consistent with the average real rate of return on US Treasury 1-yearbills), and the value of � = 0:025 implies an annual depreciation rate of 10 percent. The values of� and A1 are commonly used in calibration exercises (e.g., see Mulligan and Sala-i-Martin, 1993).

The choice of values B2 and k12 is rather arbitrary. There is no division in the data betweenindustries that produce capital whose quality augments labor (H-type capital in the model) andcapital whose quality augments capital itself (K-type capital). We thus have no direct measureof total factor productivity (TFP) in the H industry. The model is a very simple construct, con-venient for exploring new dimensions of growth theory but not realistic enough to make believableestimation of B2 by moment-matching. We therefore try di¤erent values of B2 to generate theannual growth rate in the presence of trade in the range of 2� 3 percent. For the given values ofother parameters the values of B2 consistent with this rate of growth are 0:17�0:2. Note that thisdoes not necessarily imply that B2 cannot take larger values. We can �nd another combinationof the parameter values that will generate annual growth rate in the acceptable range for highervalues of B2. Similarly, we have no direct measure of k12, making it impossible to know reasonablevalues to choose, so again we explore the model�s behavior for di¤erent values of k12. Note that B2and k12 are very di¤erent in meaning. B2 is a parameter of the model, capturing elements of theproduction technology. In contrast, k12 is the ratio of two endogenous variables. Its initial valuere�ects how close the initial value of one type of capital (K) in the �rst economy is compared tothe other type of capital (H) in the second economy. It is di¢ cult to know what is a reasonablerange of values for such a ratio. Consider the related ratio of K1 to H1. Would we expect thatto be large or small? Recall that we are thinking of H as types of capital that have embeddedin them technical advances that augment labor, whereas K is capital whose embodied technicaladvances augment K itself. A bit of re�ection suggests that a great deal of physical capital, suchas machinery and other producer durables, seems to be like H. Structures, such as factories andstorage sheds, seem more like K-type capital. In the US, the value of producer durables andstructures is about the same, suggesting that the ratio of K1 to H1 is about 1. Because we areassuming that country 1 and country 2 are large relative to each other, we would expect k12, whichis the ratio of K1 to H2, to be the same order of magnitude.

For the values of B2 in the range of 0:17 � 0:2 we tried di¤erent values of k12 to study thedynamic behavior of the system. There are three distinctive patterns in the transition behavior ofthe system. First, the dynamic system can yield complex eigenvalues with oscillatory convergenceto the BGP. Second, the system can also produce monotonic convergence to the BGP. Finally,

23

there is also some asymmetry in the pattern of convergence between countries specializing in K-type capital and H-type capital which arises from the fact that the good used as K-type capitalalso is the good used for consumption.

It turns out that the values of k12 equal to 5 and higher yield real eigenvalues and overallmonotonic approach of the variables to their BGP values, whereas the values of k12 in the rangecloser to 1 yields complex eigenvalues with oscillatory convergence to the BGP. We just arguedthat values of k12 as high as 5 or 10 are unlikely to be consistent with our assumption of twolarge economies. On the other side, very low values of k12 yield qualitative results similar to thosefor k12 = 5 and above, so it seems that reasonable values of k12 are consistent with oscillatorybehavior. The model thus suggests that cyclicality is the expected pattern of dynamic adjustmentof the countries to the BGP. We will see more cyclicality in the pattern of the adjustment of thetwo countries in the discussion of case 4, which we take up next.

4.4 Case 4: k1 > k�1, k2 > k�2

In this �nal case, both countries have relatively too much K. The set of dynamic equations forcountry 1 are again given by (31), (30) and (29). Country 2 sets investment in K to 0. Country2 imports Y from country 1 and uses all of it for consumption. The paths of C2, K2 and H2 aredetermined by the following set of the equations:

_C2C2

=1

�

�(1� �2)B2k

�22 +

_p

p� � � �

�(35)

K2t = K20e��t (36)

_H2

H2= B2k

�22 � c2k2

p� � (37)

Now the growth rates of country 1 and country both depend on the transitional behavior of theworld relative price. As in the previous case, we use the balanced trade condition to solve for thegrowth rate of the world relative price, obtaining

_p

p=

�A1k�1�11

c2k21 + c1

�(1� �1)(A1k�11 � c1k1)

p� ��� c1c2k21 + c1

�(1� �1)A1k�11

p� � � �

�� c2k21c2k21 + c1

�(1� �2)B2k

�22 � � � �

�where k21 � K2=K1. Note that this ratio is not the same as the ratio in case 3 above. However,like k12, it remains constant along the transition path for the same reasons. As in the previouscase, the transitional behavior of this world economy is described by a �ve dimensional system:266664

_p=p_c1=c1_k1=k1_c2=c2_k2=k2

377775 =266664d11 d12 d13 d14 d15d21 d22 d23 d24 d25d31 d32 d33 0 0d41 d42 d43 d44 d45d51 0 0 d54 d55

377775266664

p� p�c1 � c�1k1 � k�1c2 � c�2k2 � k�2

377775where the elements dij again are combinations of the parameters �1, �2, A1, B2, �, �, �, the BGPvalues of k1and k2, and the initial value k21 (which is constant on the transition path).13 Thissystem can be written as _z = Dzt, where _z is the �ve-dimensional vector of the growth rates of thevariables p, c1, k1, c2 and k2 and where zt is a vector of deviations of the variables from their steady

13See Yenokyan (2010) for the elements of matrix D.

24

state values. Again the high dimensionality of the system does not allow an analytical solution, sowe resort to a numerical exploration.

We use the same initial values of �1, �2, �, �, �, and A1 as in case 3: �1 = 0:3, �2 = 0:25, � = 2,� = 0:0025, � = 0:025, and A1 = 1. Here as well we keep the values of B2 in the range 0:17� 0:2and try di¤erent values of k21. Values of k21 at all near 1 produce real eigenvalues and relativelymonotonic convergence. However, the values of k21 higher than 2 generates oscillatory behavior.These results suggest that oscillatory behavior again is to be expected along the transition path,as in case 3 above.

5 Conclusion

We have seen that trade in goods can raise the growth rates of both trading partners throughcomparative advantage without there being any scale e¤ects, technology transfer, research anddevelopment, or international investment. Comparative advantage determines the pattern oftrade, that is, which good will be produced in which country. When a certain condition on themodel parameters is met, the world achieves an interior solution in which a world BGP exists, isunique, and is globally asymptotically stable. When the condition is not met, the world achievesa corner solution in which growth rates are not equalized. Consequently, in contrast to Acemogluand Ventura (2002), trade need not generate a stable world income distribution. In the interiorsolution, trade raises the growth rates of both countries. In the corner solution, trade raises thegrowth rate of the technologically smaller country but still leaves it below the growth rate of thetechnologically larger country. Trade in factors has some of the same e¤ect as technology transferand in one important special case has exactly the same e¤ect as technology transfer. We thus havethe major implications that (1) trade generally increases growth rates, (2) trade need not increasea given country�s growth rate and need not lead to growth convergence, and (3) trade can leadto growth outcomes that are equivalent to what would emerge from technology transfer. Thesee¤ects of trade on growth mean that the use of closed-economy models to analyze cross-countrydata is likely to be misleading. We also show that trade in goods that are not factors of productiondoes not a¤ect a country�s growth rate. In particular, what determines whether trade increases acountry�s growth rate or leaves it unchanged is the type of good that the country imports. If theimported good is a factor of production, trade will raise the country�s growth rate. Otherwise,trade leaves the growth rate una¤ected. Factor price equalization holds in the interior but notin the corners. The conditions for factor price equalization in this dynamic Ricardian modelare exactly the opposite from those required in the standard Hecksher-Ohlin framework. Herewe require that both countries specialize in a single good, whereas in the H-O framework bothcountries must not specialize. The unifying principle is that trade will equalize factor prices ifit leads to e¤ective equalization of technology. Neither the Stolper-Samuelson theorem nor theRybczynski theorem holds in this kind of model.

The study of the transitional dynamics reveals that there can be four scenarios describing thedynamic behavior of two trading economies, which are large relative to each other. In two of thosecases countries don�t trade along the transition. Deviations of the factor ratios from their BGPvalues lead to the situation where countries accumulate the factor of production they are lackingbefore opening to trade on the BGP. In the remaining two cases countries trade not only on theBGP but also along the transition. The evolution of the world relative price in the presence oftrade depends on control and state variables of both countries, leading to complicated dynamicsystems describing the dynamic behavior of trading countries. The analysis suggests that theBGP is saddle-path stable and that transition dynamics may be oscillatory.

25

References

References

[1] Acemoglu, D. and J. Ventura, �The World Income Distribution,�Quarterly Journal of Eco-nomics 117, May 2002, pp. 659-694.

[2] Aghion, P., and P. Howitt, �Growth with quality-improving innovations: an integrated frame-work,� in P. Aghion and S. Durlauf (eds.), Handbook of Economic Growth, North-Holland,Amsterdam, 2005.

[3] Alcala, F and A. Ciccone, �Trade and Productivity,�Quarterly Journal of Economics 119,May 2004, pp.613-646.

[4] Arabshahi, M., �Trade, Growth and Development�Dissertation, North Carolina State Uni-versity, August 2007.

[5] Backus, D., P. Kehoe, and T. Kehoe,�In Search of Scale E¤ects in Trade and Growth,�Journalof Economic Theory 58, December 1992, pp. 377-409

[6] Barro, R. J., and X. Sala-i-Martin,�Technological Di¤usion, Convergence, and Growth,�Journal of Economic Growth 2, March 1997, pp. 1-26.

[7] Barro, R. J., and X. Sala-i-Martin , Economic Growth, 2nd edition, MIT Press, Cambridge,2004.

[8] Bond, E. and K. Trask, �Trade and Growth with Endogenous Human and Physical CapitalAccumulation,�Growth and International Trade, Bjarne S. Jensen and Kar-yiu Wong (eds.),University of Michigan Press, 1997.

[9] Bond, E., Trask, K., and P. Wang, �Factor Accumulation and Trade: Dynamic ComparativeAdvantage with Endogenous Physical and Human Capital,� International Economic Review44, August 2003, pp. 1041-1060.

[10] Bond, E., Wang, P., and C. Yip, �A General Two Sector Model of Endogenous Growthwith Human and Physical Capital: Balanced Growth and Transitional Dynamics,�Journal ofEconomic Theory 68, January 1996, pp. 149-173.

[11] Coe, D. T., and E. Helpman, �International R&D Spillovers,�European Economic Review 39,May 1995, pp. 850-887.

[12] Coe, D. T., E. Helpman, and A. W. Ho¤maister, �International R&D Spillovers and Institu-tions,�National Bureau of Economic Research working paper 14069, June 2008.

[13] Connolly, M., �North-South Technological Di¤usion: A New Case for Dynamic Gains fromTrade,�Duke University Economics Department working paper #99-08, 2000.

[14] Faiq, M., �A Simple Economy with Human Capital: Transitional Dynamics, TechnologyShocks, and Fiscal Policies,�Journal of Macroeconomics 17, 1995, pp.421-446.

[15] Farmer, R., and A. Lahiri, �A Two-Country Model of Economic Growth,�Review of EconomicDynamics 8, January 2005, pp. 68-88.

[16] Farmer, R., and A. Lahiri, �Economic Growth in an Interdependent World Economy,�Eco-nomic Journal 116, October 2006, pp. 969-990.

26

[17] Feenstra, R. C., �Trade and Uneven Growth,�Journal of Development Economics 49, April1996, pp. 229-256.

[18] Galor, O., and A. Mountford, �Trade and the Great Divergence: The Family Connection,�American Economic Review 96, May 2006, pp.299-303.

[19] Gort, M., J. Greenwood, and P. Rupert, �Measuring the Rate of Technological Progress inStructures,�Review of Economic Dynamics 2, January 1999, pp. 207-230.

[20] Grossman, G. M., and E. H. Helpman, �Comparative Advantage and Long-Run Growth,�American Economic Review 80, September 1990, pp. 796-815.

[21] Grossman, G. M., and E. H. Helpman, Innovation and Growth in the Global Economy, MITPress, 1991, Cambridge MA.

[22] Howitt, P., �Steady Endogenous Growth with Population and R & D Inputs Growing,�Journal of Political Economy 107, August 1999, pp.715-730.

[23] Howitt, P., �Endogenous Growth and Cross-Country Income Di¤erences,� American Eco-nomic Review 90, September 2000, pp. 829-846.

[24] Hu, Y. M. Kemp, and K. Shimomura, �A Two-Country Dynamic Heckscher-Ohlin Model withPhysical and Human Capital Accumulation�, Economic Theory 41, October 2009, pp. 67-84.

[25] Lucas, R.E. Jr., �On the Mechanics of Economic Development,� Journal of Monetary Eco-nomics 22, July 1988, pp.3-42.

[26] Jones, C.I. �R&D-Based Models of Economic Growth,� Journal of Political Economy 103,August 1995, pp.758-784.

[27] Jones, L., and R. Manuelli �A Convex Model of Equilibrium Model: Theory and PolicyImplications,�Journal of Political Economy 98, October 1990, pp.1008-1038.

[28] Jones, R. W. �The Structure of Simple General Equilibrium Models,� Journal of PoliticalEconomy 73, December 1965, pp.557-572.

[29] Jones, R., �Globalization and the Theory of Input Trade,�Ohlin Lectures, MIT Press, 2000,Cambridge MA.

[30] Keller, W. �Are International R&D Spillovers Trade-Related? Analyzing Spillovers AmongRandomly Matched Trade Partners,� European Economic Review 42, September 1998,pp.1469-1481.

[31] Keller, W., and S. R. Yeaple �Multinational Enterprises, International Trade, and ProductivityGrowth: Firm-Level Evidence from the United States,�Review of Economics and Statistics91, November 2009, pp.821�831.

[32] MacDonald, G. and J. R. Markusen �A Rehabilitation of Absolute Advantage,� Journal ofPolitical Economy 93, April 1985, pp.277-297.

[33] Maddison, A., The World Economy: A Millennial Perspective, Development Centre Studies,OECD, 2001.

[34] Mino, K. �Analysis of a Two-Sector Model of Endogenous Growth with Capital Income Tax-ation,�International Economic Review 37, February 1996, pp.227-251.

[35] Mulligan, C. B. and X. Sala-i-Martin, �Transitional Dynamics in Two-Sector Models of En-dogenous Growth,�Quarterly Journal of Economics 108, August 1993, pp.739-773.

27

[36] Peretto, P., �Technological Change, Market Rivalry, and the Evolution of the Capitalist Engineof Growth,�Journal of Economic Growth March 1998, pp.53-80.

[37] Peretto, P., �Cost Reduction, Entry, and the Interdependence of Market Structure and Eco-nomic Growth,�Journal of Monetary Economics 43, February 1999, pp.173-195.

[38] Peretto, P., �Corporate Taxes, Growth and Welfare in a Schumpeterian Economy,�Journalof Economic Theory 137, 2007, pp.353-382.

[39] Peretto, P. and C. Smulders, �Technological Distance, Growth and Scale E¤ects,�EconomicJournal 112, July 2002, pp.603-624.

[40] Phelps, E. S., Golden Rules of Economic Growth: Studies of E¢ cient and Optimal Investment,W. W Norton, 1966, New York NY.

[41] Redding, S., �Dynamic Comparative Advantage and the Welfare E¤ects of Trade,�OxfordEconomic Papers 51, January 1999, pp. 15-39.

[42] Rivera-Batiz, L. A., and P. M. Romer, �Economic Integration and Endogenous Growth,�Quarterly Journal of Economics 106, May 1991, pp. 531-555.

[43] Romer, P. M., �Increasing Returns and Long-Run Growth,�Journal of Political Economy 94,October 1986, pp.1002-1037.

[44] Stokey, N. L. and S. Rebelo, �Growth E¤ects of Flat-Rate Taxes,�Journal of Political Eco-nomy 103, June 1995, pp.519-550.

[45] Taylor, M. S., � �Quality Ladders and Ricardian Trade,�Journal of International Economics34, May 1993, pp. 225-243.

[46] Uzawa, H., �Optimal Technical Change in an Aggregative Model of Economic Growth,� In-ternational Economic Review 6, January 1965, pp.18-31.

[47] Ventura, J., �Growth and Interdependence,�Quarterly Journal of Economics 112, February1997, pp.57-84.

[48] Wacziarg, R., and K. H. Welch, �Trade Liberalization and Growth: New Evidence,�WorldBank Economic Review 22, Issue 2, 2008, pp. 187-231.

[49] Yenokyan, K., �Fiscal Policy, Trade and Growth: A Dynamic Comparative Advantage Ap-proach,�Dissertation, North Carolina State University, August 2010.

[50] Young, A., "Learning By Doing and the Dynamic E¤ects of International Trade," QuarterlyJournal of Economics 106, May 1991, pp. 369-405

28

Figure 1: Phase diagram - Case 1

Figure 2: Price dynamics - Case 1

29

Figure 3: Price dynamics, case 2

Ta b le 1

R a t e s o f G r ow th o f G D P p e r C a p i t a

( a n nu a l av e r a g e c om p o u n d g r ow th r a t e s , p e r c e n t a g e p o in t s )

Ye a r s

R e g io n 1 0 0 0 - 1 5 0 0 1 5 0 0 - 1 8 2 0 1 8 2 0 - 7 0 1 8 7 0�1 9 1 3 1 9 1 3 - 5 0 1 9 5 0 - 7 3 1 9 7 3 - 9 8

W . E u r o p e 0 .1 3 0 .1 5 0 .9 5 1 .3 2 0 .7 6 4 .0 8 1 .7 8

U S 0 .3 6 1 .3 4 1 .8 2 1 .6 1 2 .4 5 1 .9 9

J a p a n 0 .0 3 0 .0 9 0 .1 9 1 .4 8 0 .8 9 8 .0 5 2 .3 4

A s ia / J a p a n 0 .0 5 0 .0 0 - 0 .1 1 0 .3 8 - 0 .0 2 2 .9 2 3 .5 4

A f r i c a - 0 .0 1 0 .0 1 0 .1 2 0 .6 4 1 .0 2 2 .0 7 0 .0 1

S o u r c e : M a d d i s o n ( 2 0 0 1 ) , Ta b le B -2 2 .

30

Appendix

A Solution with Trade

A.1 Individual Country Solutions

With trade, country i�s Hamiltonian is

Vi =C1��i

1� � e��t + �i

�Ai(viKi)

�i(uiHi)1��i � Ci � �Ki �Xi

�+ (38)

i

�Bi((1� vi)Ki)

�i((1� ui)Hi)1��i � �Hi +

Xi

p

�where � and are the costate variables. The necessary conditions are

_�i = ��i

"�iviAi

�viKi

uiHi

��i�1� �#� iBi�i(1� vi)

�(1� vi)Ki

(1� ui)Hi

��i�1(39)

_ i = ��i(1� �i)uiAi�viKi

uiHi

��i� i

�Bi(1� �i)(1� ui)

�(1� vi)Ki

(1� ui)Hi

��i� ��

(40)

@Vi@Ci

= C��i e��t � �i = 0 (41)

@Vi@vi

= �iAi�iKi

�viKi

uiHi

��i�1� iBi�iKi

�(1� vi)Ki

(1� ui)Hi

��i�1= 0 (42)

@Vi@ui

= �i(1� �i)AiHi

�viKi

uiHi

�� iBi(1� �i)Hi

�(1� vi)Ki

(1� ui)Hi

��i�1= 0 (43)

@Vi@Xi

= ��i + ip= 0 (44)

plus initial and transversality conditions, which are unneeded in what follows.

A.2 Balanced growth path with trade

Knowing that country 1 specializes in Y CK allows us to write a simpli�ed maximization problemfor it, conditional on the fact that it produces no Y H . Equations (38) - (44) still characterize theproblem, but now we set u = v = 1. The Hamiltonian reduces to

V1 =C1��1

1� � e��t + �1

�A1K

�11 H1��1

1 � C1 � �K1 �X1

�(45)

+ 1

�X1

p� �H1

�and the necessary conditions become

_�1 = ��1

"�1A1

�K1

H1

��1�1� �#

(46)

_ 1 = ��1(1� �1)A1�K1

H1

��1+ 1� (47)

31

@V1@C1

= C��1 e��t � �1 = 0 (48)

@V1@X1

= ��1 + 1p= 0 (49)

There no longer are �rst-order conditions for v and u because they already have been set to one.The �rst-order conditions for C and X are unchanged from the unconditional problem, but the�rst-order condition for X now holds with equality. Having accepted the world price p and agreedto specialize in producing Y CK , country 1 now chooses � and to satisfy (49) exactly.

Di¤erentiating (48) with respect to time, dividing both sides by �, and manipulating yields thegrowth equation for consumption

C1 = �1

�

_�1�1+ �

!(50)

The same kind of manipulations as for the autarkic model show that the growth rates of Y CK , K,and Y all equal the growth rate of consumption; as before, we denote this common growth rate . We obtain the growth rate of �1 from the remaining necessary conditions and the requirementsof balanced growth. Because equation (49) now always holds, we have p = 1=�1. Trade balanceconstrains p to lie in the closed interval [p2; p1], and balanced growth requires that everythingthat grows must do so at a constant rate. The only growth rate for p consistent with both theserequirements is zero, so p must be constant along the balanced growth path. The ratio 1=�1therefore also is constant, implying that the growth rates of �1 and 1 are equal. We can use thisfact together with (46) and (47) to solve for K1=H1, obtaining

K1

H1= p

�11� �1

(51)

We then substitute this expression into (46) and divide by �1 to obtain the growth rate for �1:

_�1�1= � � p�1�1��11 (1� �1)1��1A1 (52)

Finally, substituting this solution into (50) gives us what we are after, the growth rate of country1 in the presence of trade:

1;T =1

�

"A1�

�11 (1� �1)1��1

�1

p

�1��1� � � �

#

where the subscript T indicates that this growth rate pertains when country 1 trades.

Country 2�s growth rate is found the same way. Country 2 produces no Y CK , only Y H , so itsHamiltonian is

V2 =C1��2

1� � e��t + �2 [�C2 � �K2 +X2] (53)

+ 2

�B2(K2)

�2(H2)1��2 � �H2 +

X2

p

�with corresponding necessary conditions. Going through the same steps as for country 1 yields thegrowth rate

2;T =1

�

�B2�

�22 (1� �2)1��2p�2 � � � �

�(54)

32

B Factor Price Equalization

B.1 Interior

In country 1 only good Y CK is produced with the production function

Y1 = A1K�11 H1��1

1

The marginal product of K-type capital is:

rK1 =@Y1@K1

= �1A1

�K1

H1

��1�1On the BGP in the presence of trade the ratio of K to H in country 1 is

K1

H1= p

�11� �1

where p is the world price, given by equation (14) in the main text. Substituting the expressionfor p into the solution for the K=H ratio and then into the above expression for marginal product,we get

rK1 =

A�21 �

�1�21 (1� �1)�2(1��1)

B�1�12 ��2(�1�1)2 (1� �2)

(1��2)(�1�1)

! 11��1+�2

In country 2 only good Y H is produced with the production function

_H + �H = B2K�22 H

1��22

The marginal product of K in country 2 is

rK2 =@�_H + �H

�@K

= �2B2

�K2

H2

��2�1The K=H ratio in country 2 is

K2

H2= p

�21� �2

Substituting into the expression for the marginal product and using the expression for p fromequation (14) gives

rK2 =

A�2�11 �

�1(�2�1)1 (1� �1)(�2�1)(1��1)

B�1�22 ��2�1�2�22 (1� �2)

(1��2)(�1�2)

! 11��1+�2

It then is straightforward to show that

rK1 = prK2

Similar steps show that

rH1 =

A1+�21 �

�1(1+�2)1 (1� �1)(1+�2)(1��1)

B�1�12 ��2�12 (1� �2)

(1��2)�1

! 11��1+�2

rH2 =

A�21 �

�1�21 (1� �1)�2(1��1)

B�1�12 ��2(�1�1)2 (1� �2)

(1��2)(�1�1)

! 11��1+�2

so thatrH1 = prH2

33

B.2 Corner

In the corner case, country 1 produces both goods. The rates of return are derived by the samesteps as in the interior, giving

rYK1 =

A�11 �

�1�11 (1� �1)�1(1��1)

B�1�11 ��1(�1�1)1 (1� �1)

(1��1)(�1�1)

! 11��1+�1

rYH1 =

A1+�11 �

�1(1+�1)1 (1� �1)(1+�1)(1��1)

B�11 ��1�11 (1� �1)

(1��1)�1

! 11��1+�1

rHK1 =

A�1�11 �

�1(�1�1)1 (1� �1)(�1�1)(1��1)

B�1�21 ��1(�1�1)1 (1� �1)

(1��1)(�1�2)

! 11��1+�1

rHH1 =

A�11 �

�1�11 (1� �1)�1(1��1)

B�1�11 ��1(�1�1)1 (1� �1)

(1��1)(�1�1)

! 11��1+�1

rK2 = B2��22 (1� �2)

1��2

A�2�11 �

�1(�2�1)1 (1� �1)(�2�1)(1��1)

B�2�21 �

�1(�2�1)1 (1� �1)

(1��1)(�2�2)

! 11��1+�2

rH2 = B2��22 (1� �2)

1��2

A�21 �

�1�21 (1� �1)�2(1��1)

B�21 �

�1�21 (1� �1)

(1��1)�2

! 11��1+�2

With these expressions, it is straightforward to show that

rK2 6= p1rYK1

rH2 6= p1rYH1

where p1 is the world price in the corner, given by equation (11) in the main text.

34